A major source of Mr. Boehner’s limitations as speaker is simple math. Republicans control 232 seats, and 218 are required for a majority when there are no vacancies, meaning that in most instances, he can afford to lose only 14 Republican votes.

From The New York Times:

After House Republicans began their closed-door meeting on Tuesday with an impromptu rendition of “Amazing Grace,” Speaker John A. Boehner stood up with what he hoped would be an equally inspiring message.

The Senate was trying to steamroll the House with a new plan to reopen the government and lift the debt ceiling, leaving his team with only two options: “We can wait on the Senate for four days and accept whatever they give us,” he told them. “Or we can go on offense and force their hand.”

It was yet another moment of decision for Mr. Boehner, who finally finds himself at the crossroads he has been marching toward for weeks: an imminent financial default on the one hand, and on the other an unyielding conservative rank and file that persists with the futile effort to take down President Obama’s health care law even if they also take down the speaker in the process.

While his colleagues sang about how what once was lost had now been found, Mr. Boehner did not tell them a more dispiriting truth: With less than 48 hours left before the nation is set to exhaust its authority to borrow money, he and his lieutenants were running out of ideas — a fact made starkly evident by the mad and fruitless scramble on Tuesday to come up with a measure that could win enough support from his members. Around 7 p.m., he sent the House home and canceled all votes for the day.

“He’s herding cats,” said Representative Tom Price, Republican of Georgia.

Mr. Boehner initially tried to unite his conference around a plan that had a little bit for everyone. For his hard-line conservative members, Mr. Boehner’s proposal would have eliminated government contributions for the purchase of health insurance on the new exchanges for lawmakers, White House officials and their staffs, as well as forbidden the Treasury Department to use “extraordinary measures” to extend its borrowing capabilities. For his more moderate members, Mr. Boehner offered a simple appeal — his plan would have reopened the government through Dec. 15, and extended the nation’s borrowing authority through Feb. 7.

But conservatives and their advocacy groups balked, and Mr. Boehner was forced to set his plan aside.

“This is about as challenging as it can be, in terms of trying to figure out where the effective agreement is going to be to move forward and how do you get to that point where everybody looks at that and says, ‘Well, that isn’t everything but that’s a reasonable way forward,’ ” said David Winston, a Republican strategist and close adviser to Mr. Boehner.

Almost every day since the government shutdown began on Oct. 1, Mr. Boehner and his besieged band of House Republicans have trotted out a new gambit to try to outmaneuver Senate Democrats and the president. The public was upset over closed national monuments? Pass legislation reopening them. Vital health and safety services had no staff to carry out their functions? Appropriate money to finance them.

All the while, Mr. Boehner, of Ohio, tended carefully to the members of his fractious conference. He joined them Friday night for P. F. Chang’s Chinese takeout in the Capitol. After an appearance on the Sunday talk shows, he shopped at his local Harris Teeter and invited some lawmakers to his Capitol Hill town house for burgers, manning the grill himself. He assumed the role of battlefield commander, admonishing the president in public in the sternest of terms.

Mr. Boehner has also sought solace in his routines. He still takes long walks in the morning before heading to his office in the Capitol, and often eats breakfast alone at the counter of Pete’s Diner not far from his house. This month, shortly after the shutdown began, Mr. Boehner huddled in his Capitol suite with a group of allies and loyal members — Representatives Tom Cole of Oklahoma, John Kline of Minnesota and Pat Tiberi of Ohio, among others — known as Team Boehner.

The strategy has largely worked, so far satisfying the most unmanageable of Mr. Boehner’s restive conservative members. But the next 48 hours may prove the biggest test of Mr. Boehner’s career and determine his future at the top of House Republicans. “It’s tough,” said Senator Saxby Chambliss, the Georgia Republican who is one of the speaker’s closest friends in Congress. “Trying to find 218 votes is never easy. But right now, it’s harder than ever.”

Senator Lindsey Graham, Republican of South Carolina, who as a House member in the late 1990s was part of a successful coup to unseat Speaker Newt Gingrich, described Mr. Boehner’s tenuous state more bluntly, worrying aloud that if Mr. Boehner is unable to pass some proposal out of the House, the speaker may find himself in “a compromised situation.”

“I was involved in taking one speaker down,” Mr. Graham said. “I’d like to be involved in keeping this speaker, because quite frankly I think he deserves it.”

A major source of Mr. Boehner’s limitations as speaker is simple math. Republicans control 232 seats, and 218 are required for a majority when there are no vacancies, meaning that in most instances, he can afford to lose only 14 Republican votes. For much of the 113th Congress, that has meant that the only bills guaranteed to pass are simple, nonpartisan matters like renaming a federal building or ones that strongly reflect conservative policies like the ban on abortion 20 weeks after fertilization.

Because Mr. Boehner can almost never guarantee Senate leaders that his conference will vote with him, he has greatly diminished bargaining power in the current fight as his members pass one dead-end bill after another. “There’s a silent majority in the conference,” said Representative Devin Nunes, Republican of California. “We have too many guys who don’t understand that without 218 you have no leverage. At the end of the day, that’s the problem.”

That has not stopped Mr. Boehner from trying. He has been in near constant contact with Republican leaders in the Senate, who have conveyed to Mr. Boehner what they believe their moderate Democratic counterparts might be willing to accept. He has also been carefully soliciting the views of his own members, bringing in at least two small groups of moderate legislators for private meetings, and buttonholing other colleagues on the House floor and in the cloakroom.

“He is a really good listener,” said Representative James Lankford of Oklahoma, the chairman of the House Republican Policy Committee. “So he’s constantly trying to figure out, is this one person or is this a group that thinks this? And how large is that group? I just think he has a really unique skill of being able to hear people and where they are.”

Mr. Boehner may be unable to corral his maverick members, but they still rely on him heavily to bring in money for their re-elections: On Tuesday, in a telling reminder of his big-dollar prowess, Mr. Boehner’s campaign committee filed its third-quarter fund-raising report.

Boehner-affiliated committees transferred $4.1 million to the National Republican Congressional Committee during the third quarter, making him its largest source of member money, by far.